Statistical mechanics of gravitating systems: An Overview
T. Padmanabhan
备注：
Based on the Lecture Course given at the Les Houches Summer School on `Long Range Interacting Systems', 4-29 August 2008; 25 pages; 5 figures; ver 2 typo in author's name corrected and one reference added
I review several issues related to statistical description of gravitating systems in both static and expanding backgrounds. After briefly reviewing the results for the static background, I concentrate on gravitational clustering of collisionless particles in an expanding universe. In particular, I describe (a) how the non linear mode-mode coupling transfers power from one scale to another in the Fourier space if the initial power spectrum is sharply peaked at a given scale and (b) the asymptotic characteristics of gravitational clustering which are independent of the initial conditions. Numerical simulations as well as analytic work shows that power transfer leads to a universal power spectrum at late times, somewhat reminiscent of the existence of Kolmogorov spectrum in fluid turbulence.
The self-gravitating systems are formed by particles interacting through gravity. They describe structure formation in the universe. As a consequence of the long range interaction of gravity, they are inhomogeneous even at thermal equilibrium. We study the self-gravitating systems with several kinds of particles and the self-gravitating systems in the presence of the cosmological constant $ Lambda$. We formulate the statistical mechanics and the mean field approach describing the gaseous phase. We explicitely compute the density of particles and thermodynamic quantities. The presence of $ Lamb…
We study statistical mechanics of the self--gravitating system applying the cluster expansion method developed in solid state physics. By summing infinite series of diagrams, we derive a complex free energy whose imaginary part is related to the relaxation time of the system. Summation of another series yields two--point correlation function whose correlation length is essentially given by the Jeans wavelength of the system.
We discuss the differences and analogies of gravitational clustering in finite and infinite systems. The process of collective, or violent, relaxation leading to the formation of quasi-stationary states is one of the distinguished features in the dynamics of self-gravitating systems. This occurs, in different conditions, both in a finite than in an infinite system, the latter embedded in a static or in an expanding background. We then discuss, by considering some simple and paradigmatic examples, the problems related to the definition of a mean-field approach to gravitational clustering, focus…
We consider the statistical mechanics of a general relativistic one-dimensional self-gravitating system. The system consists of $N$-particles coupled to lineal gravity and can be considered as a model of $N$ relativistically interacting sheets of uniform mass. The partition function and one-particle distitrubion functions are computed to leading order in $1/c$ where $c$ is the speed of light; as $c\to\infty$ results for the non-relativistic one-dimensional self-gravitating system are recovered. We find that relativistic effects generally cause both position and momentum distribution functions …
The dynamical equations describing the evolution of a self-gravitating fluid can be rewritten in the form of a Schrodinger equation coupled to a Poisson equation determining the gravitational potential. This wave-mechanical representation allows an approach to cosmological gravitational instability that has numerous advantages over standard fluid-based methods. We explore the usefulness of the Schrodinger approach by applying it to a number of simple examples of self-gravitating systems in the weakly non-linear regime. We show that consistent description of a cold self-gravitating fluid requir…
By making an intuitive choice for the single-particle density of a system of N self-gravitating particles, without any source for the radiation of energy, we have been able to calculate the binding energy of the system by treating these particles as fermions. Our expression for the ground state energy of the system shows a dependence of $N^{7/3}$ on the particle number, which is in agreement with the results obtained by other workers. We also arrive at a compact expression for the radius of a star following which we correctly reproduce the nucleon number to be found in a typical star. Using th…
This pedagogical review addresses several issues related to statistical description of gravitating systems in both static and expanding backgrounds, focusing on the latter. After briefly reviewing the results for the static background, I describe the key issues and recent progress in the context of gravitational clustering of collision-less particles in an expanding universe. The questions addressed include: (a) How does the power injected into the system at a given wave number spread to smaller and larger scales? (b) How does the power spectrum of density fluctuations behave asymptotically at…
We compute statistical equilibrium states of rotating self-gravitating systems enclosed within a box by maximizing the Fermi-Dirac entropy at fixed mass, energy and angular momentum. We increase the rotation up to the Keplerian limit and describe the flattening of the configuration until mass shedding occurs. At the maximum rotation, the system develops a cusp at the equator. We draw the equilibrium phase diagram of the rotating self-gravitating Fermi gas and discuss the structure of the caloric curve as a function of degeneracy parameter and angular velocity. We argue that systems described b…
In the framework of nonextensive statistical mechanics, the equilibrium structures of astrophysical self-gravitating systems are stellar polytropes, parameterized by the polytropic index n. By careful comparison to the structures of simulated dark-matter halos we find that the density profiles, as well as other fundamental properties, of stellar polytropes are inconsistent with simulations for any value of n. This result suggests the need to reconsider the applicability of nonextensive statistical mechanics (in its simplest form) to equilibrium self-gravitating systems.
Self gravitating systems (SGS) in the Universe are generally thought to be non-extensive, and often show long-tails in various distribution functions. In principle, these non-Boltzmann properties are naturally expected from the peculiar property of gravity, long-range and unshielded. Therefore the ordinary Boltzmann statistical mechanics would not be applicable for these self gravitating systems (SGS) in its naive form. In order to step further, we quantitatively investigate the above two properties, non-extensivity and long-tails, by explicitly introducing various models of statistical mechan…
We provide a complete picture of the self-gravitating non-relativistic gas at thermal equilibrium using Monte Carlo simulations (MC), analytic mean field methods (MF) and low density expansions. The system is shown to possess an infinite volume limit, both in the canonical (CE) and in the microcanonical ensemble (MCE) when N, V \to \infty, keeping N/ V^{1/3} fixed. We {\bf compute} the equation of state (we do not assume it as is customary), the entropy, the free energy, the chemical potential, the specific heats, the compressibilities, the speed of sound and analyze their properties, signs an…
We show that the self-gravitating gas at thermal equilibrium has an infinite volume limit in the three ensembles (GCE, CE, MCE) when (N, V) -> infty, keeping N/V^{1/3} fixed, that is, with eta = G m^2 N/[ V^{1/3} T] fixed. We develop MonteCarlo simulations, analytic mean field methods (MF) and low density expansions. We compute the equation of state and find it to be locally p(r) = T rho_V(r), that is a local ideal gas equation of state. The system is in a gaseous phase for eta < eta_T = 1.51024...and collapses into a very dense object for eta > eta_T in the CE with the pressure becoming large…
We present an analytical and numerical study of the velocity distribution function of self gravitating collisionless particles, which include dark matter and star clusters. We show that the velocity distribution derived through the Eddington's formula is identical to the analytical one derived directly from the generalized entropy of non-extensive statistical mechanics. This implies that self gravitating collisionless structures are to be described by non-extensive thermo-statistics. We identify a connection between the density slope of dark matter structures, γ, from ρ~ r^{-γ}, and the entrop…
The thermodynamic behaviour of self-gravitating $N$-body systems has been worked out by borrowing a standard method from Molecular Dynamics: the time averages of suitable quantities are numerically computed along the dynamical trajectories to yield thermodynamic observables. The link between dynamics and thermodynamics is made in the microcanonical ensemble of statistical mechanics. The dynamics of self-gravitating $N$-body systems has been computed using two different kinds of regularization of the newtonian interaction: the usual softening and a truncation of the Fourier expansion series of …
We study the statistical mechanics of the self-gravitating gas at thermal equilibrium with two kinds of particles. We start from the partition function in the canonical ensemble which we express as a functional integral over the densities of the two kinds of particles for a large number of particles. The system is shown to possess an infinite volume limit when (N_1,N_2,V)->infty, keeping N_1/V^{1/3} and N_2/V^{1/3} fixed. The saddle point approximation becomes here exact for (N_1,N_2,V)->infty.It provides a nonlinear differential equation on the particle densities. For the spherically symmetri…
Fractal structures are observed in the universe in two very different ways. Firstly, in the gas forming the cold interstellar medium in scales from 10^{-4} pc till 100 pc. Secondly, the galaxy distribution has been observed to be fractal in scales up to hundreds of Mpc. We give here a short review of the statistical mechanical (and field theoretical) approach developed by us. We consider a non-relativistic self-gravitating gas in thermal equilibrium at temperature T inside a volume V. The statistical mechanics of such system has special features and, as is known, the thermodynamical limit does…
We complete our study of the selfgravitating gas by computing the fluctuations around the saddle point solution for the three statistical ensembles. Although the saddle point is the same for the three ensembles, the fluctuations change from one ensemble to another. The zeroes of the fluctuations determinant determine the position of the critical points for each ensemble.This yields the domains of validity of the mean field approach. Only the S wave determinant exhibits critical points.Closed formulae for the S and P wave determinants are derived.The local properties of the selfgravitating gas …
We provide a complete picture to the self-gravitating non-relativistic gas at thermal equilibrium using Monte Carlo simulations, analytic mean field methods (MF) and low density expansions. The system is shown to possess an infinite volume limit in the grand canonical (GCE), canonical (CE) and microcanonical (MCE) ensembles when (N, V) -> infty, keeping N/V^{1/3} fixed. We compute the equation of state (we do not assume it as is customary in hydrodynamics), as well as the energy, free energy, entropy, chemical potential, specific heats, compressibilities and speed of sound; we analyze their pr…
Large-scale astrophysical systems are non-extensive due to their long-range force of gravity. Here we show an approach toward the statistical mechanics of such self-gravitating systems (SGS). This is a generalization of the standard statistical mechanics based on the new definition of entropy; Tsallis statistical mechanics. Developing the composition of entropy and the generalized Euler relation, we investigate the galaxy distributions in count-in-cell method. This is applied to the data of CfA II South redshift survey.
The dynamical justifications which lie at the basis of an effective Statistical Mechanics for self gravitating systems are formulated, analyzing some among the well known obstacles thought to prevent a rigorous Statistical treatment. It is shown that N-body gravitational systems satisfy a strong chaos criterion, so supporting the assumption of an increasingly uniform spreading of orbits over the constant energy surface, i.e., the asymptotic evolution towards a microcanonical distribution. We then focus on the necessary conditions for the equivalence of statistical ensembles and remark that thi…
We discuss the statistical mechanics of rotating self-gravitating systems by allowing properly for the conservation of angular momentum. We study analytically the case of slowly rotating isothermal spheres by expanding the solutions of the Boltzmann-Poisson equation in a series of Legendre polynomials, adapting the procedure introduced by Chandrasekhar (1933) for distorted polytropes. We show how the classical spiral of Lynden-Bell & Wood (1967) in the temperature-energy plane is deformed by rotation. We find that gravitational instability occurs sooner in the microcanonical ensemble and later…
We provide a complete picture to the selfgravitating non-relativistic gas at thermal equilibrium using Monte Carlo simulations, analytic mean field methods (MF) and low density expansions. The system is shown to possess an infinite volume limit in the grand canonical (GCE), canonical (CE) and microcanonical (MCE) ensembles when(N, V) --> infinity, keeping N/ V^{1/3} fixed. We compute the equation of state (we do not assume it as is customary), as well as the energy, free energy, entropy, chemical potential, specific heats, compressibi- lities and speed of sound;we analyze their properties, sig…
I review the basic ``gravitational instability'' model for the growth of structure in the expanding Universe. This model requires the existence of small initial irregularities in the density of a largely uniform universe. These grow through linear and non-linear stages to form a complex network of clusters, filaments and voids. The dynamical equations describing the evolution of a self-gravitating fluid can be rewritten in the form of a Schrodinger equation coupled to a Poisson equation determining the gravitational potential. This approach has a number of interesting features, many of which w…
We discuss the statistical mechanics of violent relaxation in stellar systems following the pioneering work of Lynden-Bell (1967). The solutions of the gravitational Vlasov-Poisson system develop finer and finer filaments so that a statistical description is appropriate to smooth out the small-scales and describe the ``coarse-grained'' dynamics. In a coarse-grained sense, the system is expected to reach an equilibrium state of a Fermi-Dirac type within a few dynamical times. We describe in detail the equilibrium phase diagram and the nature of phase transitions which occur in self-gravitating …
We found equilibrium conditions for a self-gravitating toroidal vortex by taking into account thermal pressure. These conditions are shown to significantly differ from those for a disk and a sphere. The evolution of a thin vortex turns it into a compact vortex that loses mechanical stability for low masses at a polytropic index $γ< 4/3$ but retains stability for sufficiently high masses and densities determined by the velocity circulation in the vortex.
The idea of viewing gravitation as a many body phenomenon is put forward here. Physical arguments supporting this idea are briefly reviewed. The basic mathematical object of the new gravitational mechanics is a matrix of operators. Striking similarity of the method of R-matrix (QISM) to the mathematical formulation of the new gravitational mechanics is pointed out. The s-wave difference Schrodinger equation describing a process of emission of radiation by a gravitating particle is shown to be analogous to the Baxter equation of the QISM.
